Canadian Mathematical Society Conference Proceedings Volume 00, 19xx

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Abstract. This paper is a sample prepared to illustrate the use of the American Mathematical Society’s L^{A}TEX document class amsproc and publicationspecific variants of that class.

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1. This is a numbered first-level section head

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1.1. This is a numbered second-level section head. This is an example of a numbered second-level heading.

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1.1.1. This is a numbered third-level section head. This is an example of a numbered third-level heading.

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Lemma 1.1. Let f, g ∈ A(X) and let E, F be cozero sets in X.

1. If f is E-regular and F ⊆ E, then f is F -regular.

2. If f is E-regular and F -regular, then f is E ∪ F -regular.

3. If f(x) ≥ c > 0 for all x ∈ E, then f is E-regular.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0731-1036/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

The first author was supported in part by NSF Grant #000000.

The following is an example of a proof^{1}.

Proof. Set j(ν) = max(I\a(ν)) − 1. Then we have

∑_{i∉a(ν)} t_{i} ∼ t_{j(ν)+1} = ∏_{j=0}^{j(ν)}(t_{j+1}/t_{j}).

Hence we have

\begin{aligned}∏_{ν}( ∑_{i∉a(ν)} t_{i})^{|a(ν−1)|−|a(ν)|} & ∼ ∏_{ν} ∏_{j=0}^{j(ν)}(t_{j+1}/t_{j})^{|a(ν−1)|−|a(ν)|} \\ & = ∏_{j≥0}(t_{j+1}/t_{j})^{∑_{j(ν)≥j}(|a(ν−1)|−|a(ν)|)}. \end{aligned} (1.1)

By definition, we have a(ν(j)) ⊃ c(j). Hence, |c(j)| = n − j implies (5.4). If c(j) ∉ a, a(ν(j))c(j) and hence we have (5.5).

This is an example of an ‘extract’. The magnetization M_{0} of the Ising model is related to the local state probability P (a) : M_{0} = P (1) − P (−1). The equivalences are shown in Table 1.

Table 1 \begin{aligned}& −∞ & +∞ \\ f_{+}(x, k) & e^{\sqrt{−1}kx} + s_{12}(k)e^{−\sqrt{−1}kx} & s_{11}(k)e^{\sqrt{−1}kx} \\ f_{−}(x, k) & s_{22}(k)e^{−\sqrt{−1}kx} & e^{−\sqrt{−1}kx} + s_{21}(k)e^{\sqrt{−1}kx} \end{aligned}

Definition 1.2. This is an example of a ‘definition’ element. For f ∈ A(X), we define

(1.2) Z(f) = {E ∈ Z[X] : f is E^{c}-regular}.

Remark 1.3. This is an example of a ‘remark’ element. For f ∈ A(X), we define

(1.3) Z(f) = {E ∈ Z[X] : f is E^{c}-regular}.

Example 1.4. This is an example of an ‘example’ element. For f ∈ A(X), we define

(1.4) Z(f) = {E ∈ Z[X] : f is E^{c}-regular}.

Exercise 1.5. This is an example of the xca environment. This environment is used for exercises which occur within a section.

The following is an example of a numbered list.

1Here is an example of a footnote. Notice that this footnote text is running on so that it can stand as an example of how a footnote with separate paragraphs should be written.

And here is the beginning of the second paragraph.

Figure 1. This is an example of a figure caption with text.

Figure 2

1. First item. In the case where in G there is a sequence of subgroups

G = G_{0}, G_{1}, G_{2}, …, G_{k} = e

such that each is an invariant subgroup of G_{i}.

2. Second item. Its action on an arbitrary element X = λ^{α}X_{α} has the form

[e^{α}X_{α}, X] = e^{α}λ^{β}[X_{α}X_{β}] = e^{α}c_{αβ}^{γ}λ^{β}X_{γ}, (1.5)

(a) First subitem.

−2ψ_{2}(e) = c_{αγ}^{δ}c_{βδ}^{γ}e^{α}e^{β}.

(b) Second subitem.

(i) First subsubitem. In the case where in G there is a sequence of subgroups

G = G_{0}, G_{1}, G_{2}, …, G_{k} = e

such that each subgroup G_{i+1} is an invariant subgroup of G_{i} and each quotient group G_{i+1}/G_{i} is abelian, the group G is called solvable.

(ii) Second subsubitem.

(c) Third subitem.

3. Third item. Here is an example of a cite. See [A].

Theorem 1.6. This is an example of a theorem.

Theorem 1.7 (Marcus Theorem). This is an example of a theorem with a parenthetical note in the heading.

2. Some more list types

This is an example of a bulleted list.

• J_{g} of dimension 3g − 3;

• E_{g}^{2} = {Pryms of double covers of C = with normalization of C hyperelliptic of genus g − 1} of dimension 2g;

• E_{1,g−1}^{2} = {Pryms of double covers of C = _{P^{1}}^{H} with H hyperelliptic of genus g − 2} of dimension 2g − 1; • P_{t,g−t}^{2} for 2 ≤ t ≤ g/2 = {Pryms of double covers of C = _{C′′}^{C′} with g(C′) =

t − 1 and g(C′′) = g − t − 1} of dimension 3g − 4. This is an example of a ‘description’ list. Zero case: ρ(Φ) = {0}. Rational case: ρ(Φ) ≠ {0} and ρ(Φ) is contained in a line through 0 with

rational slope. Irrational case: ρ(Φ) ≠ {0} and ρ(Φ) is contained in a line through 0 with

irrational slope.

References

[A] T. Aoki, Calcul exponentiel des opérateurs microdifferentiels d’ordre infini. I, Ann. Inst. Fourier (Grenoble) 33 (1983), 227–250.

[B] R. Brown, On a conjecture of Dirichlet, Amer. Math. Soc., Providence, RI, 1993.

[D] R. A. DeVore, Approximation of functions, Proc. Sympos. Appl. Math. vol. 36, Amer. Math. Soc., Providence, RI, 1986, pp. 34–56.

Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Current address: Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio 43403

    E-mail address: xyz@math.university.edu

Mathematical Research Section, School of Mathematical Sciences, Australian National University, Canberra ACT 2601, Australia

    E-mail address: two@maths.univ.edu.au